The current stock price of FB is $$29.$ Alex expects that in one year the stock price of

FB will be either $$40($up move$)$ or $$20($down move$).$ The exercise price of one$-$year European

put option of FB$=$$$30$ and the one$-$period risk$-$free rate rf $=2\%$ per annum $($NOT rc$).$ Also,

investors can freely trade $($either buy or short sell$)$ the GS fund, $\{200$ units of FB European put

option $+200$ shares of FB stock$\},$ and the MS fund, $\{$N shares of FB stock $+$ risk$-$free one$-$year

CD with original investment $B$\}$ in the market. That is$,$ no arbitrage opportunity exists.

$($a$)(3\%$; $3\%)$ Find the gross payoffs $($$$)$ of the GS fund in the up and down move, respectively.

$($b$)(3\%$; $3\%)$ The GS fund and MS fund have the same future payoffs in one year. Find the

values of N and B$.$

$($c$)(3\%)$ Calculate the current price $($$$)$ of European put option of FB $($per unit$).$

$($d$)(2\%)$ If the GS fund is perfectly equivalent to the MS fund, then we can see $\{200$ units of

FB European put option$\}=\{$short sell Y shares of FB stocks $+$ cash $C$\}.$ Use this idea to

calculate the equivalent percentage margin $(\%)$ for $200$ units of FB European put option.

$($Note: Percentage margin$=($equity$)/($short position$)).$